## REVIEWS OF MODERN PHYSICS, VOLUME 82, JANUARY–MARCH 2010

## Area laws for the entanglement entropy

J. Eisert

Institute of Physics and Astronomy, University of Potsdam, 14469 Potsdam, Germany; Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom;and Institute for Mathematical Sciences, Imperial College London, Exhibition Road, London SW7 2PG, United Kingdom

M. Cramer and M. B. Plenio

Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom and Institut fu?r Theoretische Physik, Albert-Einstein-Allee 11, Universitat Ulm, D-89069 Ulm, Germany, Published 4 February 2010

"Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such “area laws” for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states. ...

In classical physics concepts of entropy quantify the

extent to which we are uncertain about the exact state of

a physical system at hand or, in other words, the amount

of information that is lacking to identify the microstate

of a system from all possibilities compatible with the

macrostate of the system. If we are not quite sure what

microstate of a system to expect, notions of entropy will

reflect this lack of knowledge. Randomness, after all, is

always and necessarily related to ignorance about the

state. ... In quantum mechanics positive entropies may arise

even without an objective lack of information. ...

In contrast to thermal states this entropy does not

originate from a lack of knowledge about the microstate

of the system. Even at zero temperature we encounter a

nonzero entropy. This entropy arises because of a fundamental

property of quantum mechanics: entanglement.

This quite intriguing trait of quantum mechanics

gives rise to correlations even in situations where the

randomness cannot be traced back to a mere lack of

knowledge. The mentioned quantity, the entropy of a

subregion, is called entanglement entropy or geometric

*
*

*entropy and in quantum information entropy of entanglement,*

which represents an operationally defined entanglement

measure for pure states ...

one thinks less of detailed

properties, but is rather interested in the scaling of the

entanglement entropy when the distinguished region

grows in size. In fact, for quantum chains this scaling of

entanglement as genuine quantum correlations—a priori

very different from the scaling of two-point correlation

functions—reflects to a large extent the critical behavior

of the quantum many-body system, and shares some relationship

to conformal charges.

At first sight one might be tempted to think that the

entropy of a distinguished region I will always possess an

extensive character. Such a behavior is referred to as a

volume scaling and is observed for thermal states. Intriguingly,

for typical ground states, however, this is not

at all what one encounters: Instead, one typically finds

an area law, or an area law with a small often logarithmic

correction: This means that if one distinguishes a

region, the scaling of the entropy is merely linear in the

boundary area of the region. The entanglement entropy

is then said to fulfill an area law. It is the purpose of this

Colloquium to review studies on area laws and the scaling

of the entanglement entropy in a nontechnical manner."